i'm just trying to solidify my understanding of whats going on in the above case so i can visualise it and why it's important so please add or correct away.
Consider $(X,d)$ a metric space, then $\exists~ \text{ a complete metric space }(\hat{X},\hat{d}) $ such that $W \subset_{dense} \hat{X}$ which is isometric to X.
so in practice let $T:X \rightarrow W$ be a distance preserving bijection, then $\forall x,y \in X, ~\exists ~ T(x), T(y) \in W $ and we have $$\hat{d}(Tx,Ty) = d(x,y)$$
and $\bar{W} = \hat{X}$
So; why is this important? i'm thinking the major effect of this is that for any sequence $x_n \in_{cauchy} X$ which is divergent (by that i mean just doesnt converge), then $\exists ~T(x_n) \in_{cauchy} W$ which is convergent in $\hat{X} = \bar{W}$ since $\hat{X}$ is complete.
i'm guessing this will be utilised with the contraction mapping theorem but this is thus far the only major revelation i could imagine as to why this is overwhelmingly important... am i missing anything?
Thanks for taking the time to read this, i appreciate it.
We want to think of the completion of $X$ as a metric space $\hat{X}$ which is the space $X$ plus extra points that we add to ensure that all the Cauchy sequences of $X$ that do not have a limit in $X$, get a limit in $\hat{X}$.
We don't want to change $X$ in any way, so all points of $X$ keep their old distance. We only have to come up with distances $\hat{d}(x,y)$ when at least one of them is a "new" point.
We don't want to add unnecessary points, so any new point in $\hat{X}$ should in fact be a limit of some sequence from $X$. Note that this sequence, as seen in $X$ is in fact a Cauchy sequence (it must be Cauchy in $\hat{d}$ (convergent implies Cauchy) and so in $d$ too as the distance is the same) that does not converge in $X$ (as limits in $\hat{X}$ are unique, and we already have one among the new points.). This explains why $X$ must be dense in $\hat{X}$.
But in practice $\hat{X}$ will be a different object from the original $X$ (equivalence classes of Cauchy sequences, usually) so as long as we have an isometric dense copy of $X$ inside a space $\hat{X}$ we identify $X$ with that copy via the isometry $T$. The points in $\hat{X}\setminus T[X]$ are then the "newly added" points.
It's more of a category theory view: the $X$ and $\hat{X}$ are different objects in the category, with a "reflection morphism" (the isometric embedding) between them such that certain diagrams can be made commutative in a unique way. There isn't always a notion of "subobject" but morphisms we always have.